Not sure if you understand fully what is happening here - NonLin FEA is not straightforward and needs time/practice and some training to understand. I would recommend a course, the are many out there (e.g., by NAFEMS)
What do you mean one face shows yield, and the other 3 not, you have to be more clear than that in order for us to help you. Do you mean the tool parts upper and lower blocks (with supports). These do not have any nonlinear properties assigned to them so they are purely elastic and will never yield. Also why would you look at the tools to go into yield, normally you look on the specimen.
As I said to see if it has yielded you need to look on plastic strains! VM stress will not always tell you if it has yielded (as for the two upper lower blocks).
If we look at the parts shown below they have yield significantly even at a displacement of 4E-4 m. So they have yielded. The other parts as mentioned (blocks), do not show in the plastic strain because they are linearly elastic as explained . In the image you show, the part has failed (large plastic region), since one does not need any more additional force to compress it further.
Hope this will explain to you what is happening.
(I have a student licence hence I need to modify the model to run it, thus half symmetry and so on).